No abstract

Numerical techniques based on piecewise polynomial (that is, spline) collation at Gaussian points are exceedingly effective for the approximate solution of boundary value problems, both for ordinary differential equations and for time dependent partial differential equations. There are several widely available computer codes based on this approach, all of which have at their core a particular choice of basis representation for the piecewise polynomials used to approximate the solutions. Until recently, the most popular approach was to use a B-spline representation, but it has been shown that the B-spline basis is inferior, both in operation counts and conditioning, to a certain monomial basis, and the latter has come more into favor. In this paper, we describe a linear algebraic equations which arise in spline collocation at Gaussian points with such a monomial basis. It is shown that the new package, which implements an alternate column and row pivoting algorithm, is a distinct improvement over existing packages from the points of view of speed and storage requirements. In addition, we describe a second package, an important special case of the first, for solving the almost block diagonal systems which arise when condensation is applied to the systems arising in spline collocation at Gaussian points, and also in other methods for solving two-point boundary value problems, such as implicit Runge-Kutta methods and multiple shooting.

ABDPACK is a package of FORTRAN programs for the solution of systems of linear equations with the almost block diagonal structure arising in spline collocation at Gaussian points with monomial spline basis functions, when applied to two-point boundary value problems with separated boundary conditions. The package ABBPACK is designed to handle a subclass of such linear systems which have what may be called an almost block bidiagonal structure. Such systems result, for example, when condensation is applied to the full spline collocation linear system. This package may also be used to solve the almost block bidiagonal systems arising in multiple shooting techniques and implicit Runge-Kutta methods for solving two-point boundary value problems. The algorithms implemented in the package are based on an alternate column and row pivoting scheme which avoids most of the fill-in introduced by more commonly used techniques.

No abstract